Wire constructions of Abelian topological phases in three or more dimensions
Abstract
Coupledwire constructions have proven to be useful tools to characterize Abelian and nonAbelian topological states of matter in two spatial dimensions. In many cases, their success has been complemented by the vast arsenal of other theoretical tools available to study such systems. In three dimensions, however, much less is known about topological phases. Since the theoretical arsenal in this case is smaller, it stands to reason that wire constructions, which are based on onedimensional physics, could play a useful role in developing a greater microscopic understanding of threedimensional topological phases. In this paper, we provide a comprehensive strategy, based on the geometric arrangement of commuting projectors in the toric code, to generate and characterize coupledwire realizations of stronglyinteracting threedimensional topological phases. We show how this method can be used to construct pointlike and linelike excitations, and to determine the topological degeneracy. We also point out how, with minor modifications, the machinery already developed in two dimensions can be naturally applied to study the surface states of these systems, a fact that has implications for the study of surface topological order. Finally, we show that the strategy developed for the construction of threedimensional topological phases generalizes readily to arbitrary dimensions, vastly expanding the existing landscape of coupledwire theories. Throughout the paper, we discuss topological order in three and four dimensions as a concrete example of this approach, but the approach itself is not limited to this type of topological order.
Contents
 I Introduction

II Threedimensional wire constructions
 II.1 Decoupled wires
 II.2 Interwire couplings and criteria for producing gapped states of matter

II.3 Fractionalization
 II.3.1 Change of basis
 II.3.2 Compactification, vertex operators, and fractional charges
 II.3.3 Pointlike and linelike excitations
 II.3.4 Energetics of pointlike and linelike defects
 II.3.5 Statistics of pointlike and linelike defects
 II.3.6 Topological groundstate degeneracy
 II.3.7 Topological field theory
 II.4 Example: topological order in threedimensional space from coupled wires
 III Higherdimensional wire constructions
 IV Conclusion
 A Deconfinement of defects along the direction of a wire
 B Discrete gauge symmetry and ground state in the limit of vanishing kinetic energy
I Introduction
The experimental discovery of the integer and fractional quantum Hall effects excited enormous interest in the study of topological states of matter in two dimensional space. Strongly interacting states of matter distinguished by the presence of excitations with fractional quantum numbers or nontrivial boundary modes have attracted particular attention from theorists. Over time, a vast arsenal of theoretical tools has been developed to study such systems, from the microscopic (e.g., numerical techniques to study lattice models with topologically ordered ground states) to the macroscopic (e.g., topological quantum field theories).
Wire constructions, which were first undertaken for the integer Poilblanc et al. (1987); Yakovenko (1991); Lee (1994), and later the fractional Kane et al. (2002); Teo and Kane (2014); Mong et al. (2014); Klinovaja and Loss (2014); Meng et al. (2014); Klinovaja and Tserkovnyak (2014); Sagi and Oreg (2014); Neupert et al. (2014); Meng and Sela (2014); Klinovaja et al. (2015); Santos et al. (2015); Sagi et al. (2015), quantum Hall effect, are conveniently poised midway between these two extremes. The approach in this case is to model a topological phase by starting from an anisotropic theory of decoupled gapless quantum wires, and then introducing local couplings between the wires to produce a gapped state of matter with an isotropic lowenergy description. This approach has the virtue of yielding the edge theory, itself that of a Luttinger liquid, directly, and of providing means to construct the lowlying quasiparticle excitations of the bulk quantum liquid. Furthermore, because wire constructions make use of wellunderstood techniques in onedimensional physics, such as Abelian (or nonAbelian) bosonization, one can construct analytically tractable theories of states of matter that might not otherwise admit a controlled analytical description.
In recent years, wire constructions have also been used to study fractional topological insulators (FTIs) Neupert et al. (2014); Sagi and Oreg (2014); Santos et al. (2015) and spin liquids Meng et al. (2015); Gorohovsky et al. (2015), and also to develop an extension Neupert et al. (2014) of the tenfold way for noninteracting fermions Altland and Zirnbauer (1997); Schnyder et al. (2009); Kitaev (2009); Ryu et al. (2010) to stronglycorrelated systems.
Since the prediction Fu et al. (2007) and discovery Hsi (); Hsieh et al. (2009); Che (a) of threedimensional topological insulators (TIs), there has been a growing interest in understanding topological states of matter in three spatial dimensions. In addition to generalizing these timereversal invariant topological insulator to the stronglyinteracting regime Maciejko et al. (2010); Swingle et al. (2011); Maciejko et al. (2014), there has been an effort to derive effective field theories describing the bulk of such TIs, and to determine the bulkboundary correspondence in such theories that yields the hallmark single Dirac cone on the twodimensional surface Cho and Moore (2011); Chan et al. (2013); Tiwari et al. (2014); Ye and Gu (2015); Che (b). Further work has undertaken efforts to understand broader features of threedimensional topological states of matter, such as the statistics of pointlike and linelike excitations Wang and Levin (2014); Lin and Levin (2015). For example, it has been shown that certain threedimensional topological phases can only be distinguished by the mutual statistics among three linelike excitations Lin and Levin (2015).
Another major direction of work concerns threedimensional systems whose surfaces are themselves twodimensional topological states of matter. The simplest example of this phenomenon occurs on the surface of a TI when timereversal symmetry is locally broken by a magnetic field on the surface, in which case a halfinteger surface quantum Hall effect develops Fu and Kane (2007); Qi et al. (2008); Xu et al. (2014); Yoshimi et al. (2015). Further theoretical work has shown that generic threedimensional topological phases, including but not limited to the fermionic TI, can exhibit more exotic surface topological phases that cannot exist with the same realization of symmetries for local Hamiltonians in purely twodimensional space. This family of surface phenomena is known as surface topological order von Keyserlingk et al. (2013); Vishwanath and Senthil (2013); Wang and Senthil (2013); Wang et al. (2013); Burnell et al. (2014); Chen et al. (2014); Metlitski et al. (2015); Mross et al. (2015). Several recent works Mross et al. (2015); Sahoo et al. (2015) have approached the question of surface topological order by applying the quasionedimensional physics of wire constructions, although it appears that this approach necessitates the use of an unusual “antiferromagnetic” timereversal symmetry rather than the usual (physical) realization of reversal of time, which acts onsite. It is possible that a fully threedimensional wire construction could remedy this peculiarity, although such a description is still lacking.
Layer constructions, in which planes of twodimensional topological liquids are stacked on top of one another and coupled, were used to construct the single surface Dirac cone of the threedimensional TI Hosur et al. (2010) and to study surface topological order Jian and Qi (2014). Wire constructions of threedimensional topological states of matter have also recently been undertaken, yielding Weyl semimetals Vazifeh (2013); Meng (2015) and a class of fractional topological insulators Sagi and Oreg (2015). However, in all three cases, different methods are used to develop the wire constructions themselves, and little effort has been made to extend these constructions beyond the specific problem at hand in each example. In order to attack the most distinctive aspects of topological states of matter in three dimensions, such as surface topological order, it is therefore necessary to develop a framework that lends itself readily to a variety of approaches with minimal modifications.
In this paper, we provide a comprehensive strategy to design wire constructions of stronglyinteracting Abelian topological states of matter in three dimensions. The strategy that we present is to start with decoupled quantum wires placed on the links of a twodimensional square lattice, and then to couple the wires with manybody interactions associated with each star and plaquette of the lattice. In this way, each interaction term that couples neighboring wires can be viewed as corresponding to one of the commuting projectors that enters Kitaev’s toric code Hamiltonian Kitaev (2003). This correspondence simplifies the application of a criterion, first proposed by Haldane, to ensure that these interaction terms do not compete, and are sufficient in number to gap out all gapless modes in the array of quantum wires when periodic boundary conditions are imposed along all three spatial directions.
When all interaction terms satisfy this criterion, the Hamiltonian is frustrationfree, and taking the strongcoupling limit produces a gapped threedimensional state of matter. With this done, one can proceed to characterize this state of matter in terms of its pointlike and linelike excitations, as well as their statistics, and calculate the topological degeneracy, if any, of the groundstate manifold. The class of threedimensional models studied in this work features a topological degeneracy given by , where the integervalued matrix contains information about the mutual statistics of pointlike and linelike excitations in the theory. This is in close analogy with the matrix formalism developed for twodimensional topological states of matter Wen and Zee (1992). When periodic boundary conditions are relaxed by the presence of twodimensional terminating surfaces, we further show that gapless surface states result. One can apply the coupledwire techniques already developed in two dimensions to study the various gapped surface states that can be produced by introducing interwire hoppings or interactions on the surface, provided that the added terms are compatible with the interactions in the bulk.
In addition, we show that the above strategy for constructing threedimensional Abelian topological states of matter can be readily extended to arbitrary dimensions, vastly expanding the existing scope of the coupledwire approach. Indeed, much as it is possible to define higherdimensional versions of the toric code on hypercubic lattices (see, e.g., Ref. Mazáč and Hamma (2012)), one can arrange a set of decoupled quantum wires on a dimensional hypercubic lattice and couple them with interactions defined on stars and plaquettes of this lattice. Applying Haldane’s compatibility criterion, one can show that these interactions produce a gapped dimensional state of matter, whose excitations and topological properties can be investigated much as in the threedimensional case.
The structure of this paper is as follows. In Sec. II, we develop in detail the strategy discussed above for constructing threedimensional topological phases from coupled wires. In Sec. II.1, we establish the basic notation used to describe the array of decoupled quantum wires. In Sec. II.2, we present Haldane’s compatibility criterion and a class of manybody interactions between wires that satisfy it. (This class is mainly chosen for analytical expedience, and is not the only class of interactions that can be constructed according to our strategy.) In Sec. II.3, we show how to use the interacting arrays of quantum wires defined in Secs. II.1 and II.2 to study states of matter with fractionalized excitations. In particular, we show how to construct pointlike and linelike excitations, and determine their statistics, as well as the topological ground state degeneracy. Next, in Sec. II.4 we exemplify our strategy with perhaps the simplest type of topological order in three dimensions, namely topological order. Furthermore, we investigate the surface states of these topologicallyordered states of matter, and find that they are unstable to interwire hoppings. Additionally, a surface fractional quantum Hall effect with Hall conductivity can develop at the expense of breaking timereversal symmetry on the surface. We also discuss how these observations regarding surface states can be extended to the more general class of interwire interactions introduced in Sec. II.2.
Next, in Sec. III, we outline the generalization of our results to arbitrary dimensions. In Sec. III.1, we discuss how to define dimensional hypercubic arrays of quantum wires that are analogous to the square array of quantum wires used to construct threedimensional topological states. Then, in Sec. III.2, we generalize the results of Sec. II.2 regarding the definitions of appropriate interwire couplings and their compatibility in the strongcoupling limit. Finally, in Sec. III.3, we provide an example of this generalization by constructing topologicallyordered states of matter in four dimensions, and constructing their pointlike, linelike, and membranelike excitations, before concluding in Sec. IV.
Ii Threedimensional wire constructions
In this section, a method to construct arrays of coupled wires realizing topological phases of matter in threedimensional space is presented. We begin by defining a class of gapless theories describing decoupled wires, before moving on to a discussion of interwire couplings. In particular, we provide a set of algebraic criteria that are sufficient to determine whether the theory is gapped when periodic boundary conditions are imposed.
ii.1 Decoupled wires
We consider a twodimensional array of quantum wires, labeled by Latin indices , placed on the links of a twodimensional square lattice embedded in threedimensional Euclidean space. Each quantum wire is assumed to be gapless and nonchiral, and therefore to contain gapless degrees of freedom, labeled by Greek indices . We take the wires (of length ) to lie along the direction, and the square lattice to lie in the  plane. We will impose periodic boundary conditions in all directions (, , and ) until further notice. The set of decoupled quantum wires is described by the quadratic Lagrangian
(1a)  
where  
(1b)  
is a vector that collects the scalar fields defined in each of the wires. We use vertical bars as a visual aid to separate degrees of freedom defined in different wires. The blockdiagonal dimensional matrix  
(1c)  
where is the unit matrix of dimension and is a symmetric matrix with integer entries, yields the equaltime commutation relations  
(1d)  
We will omit the explicit time dependence of the fields from now on. Finally, the blockdiagonal matrix  
(1e)  
where the matrix is real, symmetric, and positivedefinite. The matrix is set by microscopics within each wire, and will usually be taken to be a diagonal matrix in this work. However, the matrix , which enters the commutation relations (1d), contains crucial data that define the fundamental degrees of freedom in a wire. The final data necessary to complete the definition of the theory describing the twodimensional array of decoupled quantum wires is the dimensional “chargevector”  
(1f) 
The dimensional integer vector collects the electric charges associated with the scalar fields , .
The theory defined by Eqs. (1) can be viewed as an effective lowenergy description of a twodimensional array of decoupled physical quantum wires containing fermionic or bosonic degrees of freedom.
For fermions, each wire contains flavors of chiral scalar fields and , where label right and leftmoving degrees of freedom, respectively. These fields obey the chiral equaltime commutation relations
(2a)  
and therefore, for fermions, the matrix entering Eq. (1d) is given by  
(2b)  
We further adopt the convention that the chargevector  
(2c) 
in units where the electron charge is set to unity, for a fermionic wire with channels. Treating an array of fermionic quantum wires within Abelian bosonization, as we do here, further requires the use of Klein factors, which are needed in order to assure that fermionic vertex operators (defined below) defined in different wires anticommute with one another. These Klein factors can be subsumed into the equaltime commutation relations for the scalar fields and . This can be done by integrating both sides of Eqs. (2a) over all and fixing the arbitrary constant of integration to be the Klein factor necessary to ensure the appropriate anticommutation of vertex operators. We refer the reader to the Appendix of Ref. Neupert et al. (2011) for more details on this procedure.
For bosons, each wire instead contains flavors of nonchiral scalar fields and , where label “charge” and “spin” degrees of freedom, respectively. These fields obey the equaltime commutation relations
(3a)  
so that the matrix for bosons is  
(3b)  
We take the bosonic charge vector to be  
(3c) 
in units where the electron charge is set to unity, so that the fields carry a electric charge, while is neutral. (Of course, one could define a “spin vector” analogous to that encodes the coupling to another gauge field for spin, but, for simplicity, we will work exclusively with electric charges here.)
The fundamental excitations of a fermionic or bosonic wire can be built out of the vertex operators
(4) 
for any and , where we have adopted the convention of summing over repeated indices. Any local operator acting within a single wire can be built from these vertex operators. Similarly, operators spanning multiple wires can be built by taking products of vertex operators from each constituent wire. The charges of the excitations created by these vertex operators are measured by the charge operator
(5) 
for any and , where is the length of a wire. The normalization of the charge operator is taken to be such that
(6) 
at equal times, indicating that the vertex operator carries the charge .
ii.2 Interwire couplings and criteria for producing gapped states of matter
Given the twodimensional array of decoupled and gapless quantum wires defined in Sec. II.1, we would like to devise a systematic way of introducing strong singleparticle or manybody couplings between adjacent wires in order to yield a variety of gapped topologicallynontrivial threedimensional phases of matter. Our strategy will be to extend the approach taken in Ref. Neupert et al. (2014), which considered onedimensional chains of wires, to two dimensions. We begin by adding to the quadratic Lagrangian defined in Eq. (1) a set of cosine potentials
(7) 
Here, the dimensional integer vectors encode tunneling processes between adjacent wires. This interpretation becomes transparent upon recognizing that, up to an overall phase,
(8) 
where are the vertex operators defined in Eq. (4). [We follow Ref. Neupert et al. (2014) in using the shorthand notation and in employing an appropriate pointsplitting prescription when multiplying fermionic operators.] For generic tunneling vectors , Eq. (8) describes a manybody or correlated tunneling that amounts to an interaction term in the Lagrangian
(9) 
The realvalued functions and in Eq. (7) encode the effects of disorder on the amplitude and phase of these interwire couplings.
Distinct states of matter can be realized by restricting the sum over tunneling vectors in Eq. (7) to ensure that the interaction terms (8) satisfy certain symmetries. For all examples considered in this work, we will assume that either charge or numberparity conservation holds. The former is imposed by demanding that
(10a)  
while the latter is imposed by relaxing the above requirement to  
(10b) 
For a detailed discussion of how further symmetry requirements constrain the tunneling vectors , see Ref. Neupert et al. (2014).
We are now prepared to discuss the strategy we employ to produce gapped states of matter from the above construction. We first recall that the array of decoupled quantum wires consists of gapless degrees of freedom. As noted in Ref. Haldane (1995), and later employed in Refs. Neupert et al. (2011, 2014), a single cosine term in the sum in Eq. (7) is capable of removing (i.e., gapping out) at most two of these gapless degrees of freedom from the lowenergy sector of the theory. This occurs in the limit , where the argument of the cosine term becomes pinned to its classical minimum. Therefore, in principle it takes only cosine terms to gap out all degrees of freedom in the bulk of the array of quantum wires when periodic boundary conditions are imposed. Matters are complicated somewhat by the nontrivial commutation relations (1d), which ensure that cosine terms corresponding to distinct tunneling vectors and do not commute in general. Consequently, it is possible that quantum fluctuations may lead to competition between the various cosine terms that frustrates the optimization problem of simultaneously minimizing all of these terms. However, in Ref. Haldane (1995), Haldane observed that if the criterion
(11) 
holds, then the cosine terms associated with the tunneling vectors and can be minimized independently, and therefore do not compete with one another. [Note that each tunneling vector must also satisfy Eq. (11), i.e., we require that for all .] Therefore, if one can find a “Haldane set” of linearlyindependent tunneling vectors, all of which satisfy Eq. (11), then it is possible to gap out all degrees of freedom in the array of quantum wires by adding sufficiently strong interactions of the form (7). If such a set is found, then it suffices to restrict the sum in Eq. (7) to , and to posit that all couplings are sufficiently large in magnitude to gap out all modes in the array of quantum wires.
We now present a simple geometric prescription to aid in the determination of the existence (or lack thereof) of a Haldane set for a twodimensional array of quantum wires with some set of desired symmetries. This prescription capitalizes on the fact that we have chosen all quantum wires to lie on the links of a square lattice. (In principle, this is not the only possible choice of lattice geometry, but it provides a simple way of counting degrees of freedom in any dimension, as we will see below and in Sec. III.) On a square lattice with sites, there are “stars” (centered on the vertices of the lattice) and “plaquettes” (centered on the vertices of the dual lattice), assuming that periodic boundary conditions are imposed as in Fig. 1. If we associate the tunneling vectors and with each star and plaquette , respectively, then we have a set of tunneling vectors. Since there are gapless degrees of freedom in the array of decoupled quantum wires, we can obtain the necessary number of tunneling vectors by expanding this set to include “flavors” of tunneling vectors and for each star and plaquette, respectively. We label these flavors using a teletype index . Imposing the Haldane criterion (11) on this set of tunneling vectors then yields the set of equations
(12a)  
(12b)  
(12c) 
If the above equations are satisfied, then the set of tunneling vectors is a Haldane set, and therefore capable of yielding a gapped phase in the strongcoupling limit.
We now turn to the problem of building tunneling vectors and . Enumerating all solutions to this problem for all matrices is beyond the scope of the present work. However, we will present below one way of constructing these tunneling vectors that builds in the minimal symmetries of charge and/or parity conservation [Eqs. (10)] and greatly reduces the number of equations that must be solved [relative to Eqs. (12), which contain an infinite number of linear equations in the thermodynamic limit if no additional information is provided]. In particular, if we desire charge conservation [Eq. (10a)] to hold, we may define the tunneling vectors by their nonvanishing components
(13a)  
(13b) 
where we recall that labels the quantum wires and labels the degrees of freedom within a wire. Here, , , , and are arbitrary dimensional integer vectors. The Kronecker deltas in the tunneling vector ensure that its nonzero entries are defined within the quantum wires to the north, , west of the vertex on which star is centered. The Kronecker deltas in select the quantum wires , which are defined similarly for the plaquette (see Fig. 1). With these definitions, one verifies that Eq. (10a) holds independently of the form of , , and the chargevector for a single wire.
Similarly, when we wish to impose numberparity conservation [Eq. (10b)], we may define for any and any
(14a)  
(14b) 
and verify that Eq. (10b) holds independently of the form of , , and .
Henceforth, we will focus on the chargeconserving tunneling vectors defined in Eqs. (13), as all general criteria discussed below have analogues for the parityconserving tunneling vectors defined in Eqs. (14).
The chargeconserving tunneling vectors defined in Eqs. (13) are expressed in a convenient pictorial form in Fig. 2. From this pictorial representation, it is clear that any two distinct, adjacent stars (be they of the same flavor or different flavors) share a single wire between them. The same statement holds for plaquettes. However, adjacent stars and plaquettes share two wires between them, regardless of the flavor. Therefore, one can show that Eqs. (12) are satisfied if and only if
(15a)  
(15b)  
(15c) 
for all j and and . Equations (15) are fundamental to our construction, as each solution to these equations for a given dimension of the matrix may in principle describe a distinct gapped phase of matter.
Observe that Eqs. (15) are symmetric under and . Therefore, these criteria amount to a set of linear equations in variables. This is important for two reasons. First, the number of equations does not scale with the number of quantum wires in the array. This ensures that a single solution to these equations holds for any system size when periodic boundary conditions are imposed. Second, this set of equations is underconstrained for any (i.e., there are always more variables than equations). This means that for generic matrices of fixed dimension , there is in principle more than one solution to Eqs. (15).
We aim to construct gapped states of matter that have an isotropic lowenergy description. Consequently, it is natural to demand that the tunneling vectors defined in Eqs. (13) and depicted in Fig. 2 are independent of direction. This can be achieved by imposing the additional constraints
(16a)  
and  
(16b) 
Note that Eq. (15c) is solved independently of the form of the dimensional vectors and if Eqs. (16) hold. These constraints reduce the total number of variables contained in the tunneling vectors and from to , and the number of nontrivial equations to , i.e.,
(17a)  
(17b) 
which are merely rewritings of Eqs. (15a) and (15b). With this, we have arrived at the simplest incarnation of our construction. We will henceforth assume that Eqs. (17) hold for appropriate choices of the , dimensional vectors and . However, note that Eqs. (16) are sufficient but not necessary in order to produce a state of matter that has an isotropic lowenergy description. We will therefore comment, as appropriate, on how our results below generalize to cases where and .
ii.3 Fractionalization
ii.3.1 Change of basis
In this section, we outline how to use twodimensional arrays of coupled quantum wires, like those described in the previous two sections, to study phases of matter with fractionalized excitations. To this end, let us assume that we have a Haldane set containing tunneling vectors and with defined by Eqs. (13) that satisfy (16) and the Haldane criterion (17). With these assumptions, the twodimensional array of coupled quantum wires acquires a gap in the strongcoupling limit, yielding a threedimensional gapped state of matter.
As discussed in the previous section, the phase of matter obtained in this way is a system of stronglyinteracting fermions or bosons. However, for the purposes of studying fractionalization, it is convenient to work in a basis where the “fundamental” constituents of each wire are not fermions or bosons, but (possibly fractionalized) quasiparticles. This is achieved by making the change of basis
(18a)  
(18b)  
(18c)  
(18d)  
(18e)  
where  
(18f)  
for some invertible integervalued matrix . 
This change of variables has several virtues. First, remains symmetric and integer valued. Second, remains integer valued. Third, this change of variables leaves the quantity , which enters the argument of the cosine terms in Eq. (7), invariant, i.e.,
(19) 
Thus, the linear transformation (18) does not change the character of the interaction itself, although it alters the tunneling vector and the dimensional vector of bosonic fields. Furthermore, one verifies that the linear transformation (18) does not alter the compatibility criteria (15) or the quantity that determines the presence or absence of charge or numberparity conservation.
Given the possibility of performing a change of basis of the form (18), we may now take a different approach. Instead of viewing the wire construction as a theory, with the Lagrangian (9), of scalar fields obeying the commutation relations (1d) with a matrix [Eq. (2)] for fermions or [Eq. (3)] for bosons, we may also view it as a theory, with the Lagrangian
(20) 
of scalar fields obeying the new equaltime commutation relations
(21) 
for and , which are neither fermionic nor bosonic in nature. We allow to be any symmetric, invertible, integer matrix, as long as it is related to or by a transformation of the form (18). Interactions between wires that yield a gapped state of matter can be constructed by following the procedures of the previous section. The integer tunneling vectors and obtained in this way form a Haldane set related to the Haldane set by the transformation (18). For reasons of simplicity that will become clear momentarily, we will concern ourselves in this paper primarily with the tunneling vectors and whose nonzero entries are equal to . (Of course, nothing prevents us from also considering cases where this does not hold.) The counterparts and of these tunneling vectors under the transformation (18) generically have entries with magnitude larger than 1. This fact will be of importance to us now, as we turn to the issue of compactification.
ii.3.2 Compactification, vertex operators, and fractional charges
Although the transformation (18) might appear innocuous, there is a fundamental difference between the theory with the Lagrangian defined in Eq. (20) and the original fermionic or bosonic theory with the Lagrangian defined in Eq. (9) when periodic boundary conditions are imposed in the direction (as we have assumed from the outset). In the latter theory, which is a theory of interacting electrons or bosons treated within bosonization, the traditional choice of compactification for the scalar fields with and is
(22) 
for . This choice ensures the singlevaluedness of the fermionic or bosonic vertex operators (4) under , and, in turn, that of the Lagrangian , as one can rewrite in terms of the correlated tunnelings (8), which reduce to products of these vertex operators. However, depending on the tunneling vectors , there may be other, less stringent, compactifications of these scalar fields that also render the Lagrangian singlevalued under . The parsimonious course of action is to choose the “minimal” compactification, i.e., the smallest compactification radius that still maintains the singlevaluedness of under .
If the tunneling vectors and correspond to the tunneling vectors and under the transformation (18) whose only nonzero entries are equal to , then there is a clear choice of minimal compactification. This choice can be obtained as follows. Working in the tilde basis, we can rewrite the interactions using the relation [analogous to (8)]
(23a)  
thereby implicitly defining a new set of fermionic or bosonic vertex operators,  
(23b) 
The minimal compactification is then obtained by demanding that this new set of vertex operators be single valued under . For any and , this is achieved by imposing the periodic boundary conditions
(24) 
for . Here, there is an important difference with respect to Eq. (22). Because is an integervalued matrix, is generically a rationalvalued matrix. The field is thus allowed to advance by rational (rather than integer) multiples of when the coordinate is advanced through a full period . This crucial distinction is what allows for the existence of fractionallycharged operators in the coupled wire array, as we now demonstrate.
Fractional quantum numbers appear in the wire construction because the compactification condition (24) allows for the existence of “quasiparticle” vertex operators
(25) 
for any and that are multivalued under the operation . The fact that these vertex operators generically carry fractional charges can be seen by considering the transformed charge operator
(26) 
for any and . Its normalization is here chosen such that the fermionic or bosonic vertex operators defined in Eq. (23b) have charge . Indeed, for any and , the equaltime commutator
(27) 
indicates that, since is generically a rational matrix, the quasiparticle operator generically has a rational charge. In particular, if under the transformation (18), and has at least one rational entry with magnitude smaller than 1, the operator must then carry a fractional charge.
ii.3.3 Pointlike and linelike excitations
We now outline the relationship between the quasiparticle vertex operators defined in Eq. (25) and (possibly fractionalized) excitations in the array of coupled quantum wires. In the strongcoupling limit , the compatibility criteria (15) ensure that the quantity
(28) 
where or , is pinned to a classical minimum of the corresponding cosine potential in . Following Refs. Kane et al. (2002); Teo and Kane (2014) and subsequent works, we identify excitations in the coupledwire theory with solitons that increment the “pinned field” by an integer multiple of . These excitations can therefore be viewed as living on either the stars or the plaquettes of the square lattice, rather than within the wires themselves.
We now demonstrate that products of an appropriate number of quasiparticle vertex operators of the form (25) can be used to move the soliton defects to adjacent stars and plaquettes. To see this, we write out the pinned fields explicitly for all tunneling vectors with defined on the stars ,
(29a)  
and for all tunneling vectors with defined on the plaquettes ,  
(29b) 
For any star or plaquette from the square lattice and for any , observe that, by Eq. (21), the equaltime commutators
(30) 
hold. Here, the uppercase Latin index labels the four cardinal directions. Equation (30) indicates that the pair of fields and can be viewed, up to a multiplicative constant, as canonical conjugates to the pair of fields and that enter the pair of pinned fields and , respectively. Interpreted this way, Eqs. (30) suggest that, for any , , and , the operators
(31a)  
and  
(31b) 
act on the pinned fields as [ denotes the magnitude squared of the vector ]
(32a)  
(32b)  
(32c)  
(32d) 
and [ denotes the magnitude squared of the vector ]
(33a)  
(33b)  
(33c)  
(33d) 
respectively. To verify Eqs. (32), one integrates both sides of the equalities entering Eq. (30) over the variable and uses the identity
(34) 
where is the Heaviside step function. If the underlying quantum wires in the theory are fermionic, the arbitrary integration constants above are identified with Klein factors that are necessary in order to ensure the anticommutation of fermionic vertex operators in different wires. If the underlying wires are bosonic, however, the integration constants can be set to zero.
Evidently, the operators and defined in Eqs. (31) create solitons in the pinned fields and , respectively. However, the link on star is shared with the star adjacent to along the cardinal direction , and, likewise, the link on plaquette is shared with the plaquette adjacent to along the cardinal direction . Therefore,
(35a)  
(35b)  
(35c)  
(35d) 
and
(36a)  
(36b)  
(36c)  
(36d) 
respectively. Note the sign difference with respect to Eqs. (32) and (33). This difference also stems from Fig. 2. Consequently, we interpret the operators and as creating a solitonantisoliton pair straddling the links and , respectively (see Fig. 3). By taking the derivative with respect to of Eqs. (32), (33), (35), and (36), we can interpret the operators and as creating a dipole in the soliton density across the links and , respectively. Correspondingly, the annihilation vertex operators and reverse the orientations of these dipoles in the soliton density.
The defect and antidefect created by applying one of the operators and can be propagated away from one another in the  plane by subsequent applications of the same operators on adjacent links, each of which “heal” one defect while creating another. An example of such a process is shown in Fig.